Kernel of a continuous linear transformation $T$ on a topological vector space $X$

Question

$Z$ is a closed subspace of a topological vector space $X$. Is it possible to find a continuous linear transformation $T$ from $X$ into itself, such that the $kernel(T)$ is $Z$? In the case of finite dimensional vector space, it is possible. What about infinite dimensional case?

Answer 1

This is definitely true for Hilbert spaces; consider the orthogonal projection $p_Z$ and take the identity operator minus $p_Z$.

This is also true for a Banach space $X$ and a closed, complemented subspace $Z$, that means that there exists another closed subspace $Y\subset X$ such that $X=Y+Z$ and $Y\cap Z=0$. To see this, note that any $x\in X$ is written uniquely as $x=y+z$ with $y\in Y, z\in Z$. Define $p(x)=y$. Obviously $\ker(p)=Z$ and $p$ is continuous: indeed, we can apply the closed graph theorem: if $x_n\to0$ and $p(x_n)\to y_0\in Y$, then $x_n-p(x_n)\in Z$ for all $n$, but $x_n-p(x_n)\to-y_0\in Y$, so $y_0\in Y\cap Z=0$ and we are done.

This is not true in general, not even for locally convex spaces, not even for normed spaces, not even for Banach spaces, not even for reflexive Banach spaces(!). There are some conditions you may add on $Z$ to get what you want, assuming $X$ is Banach of course; You can see some relevant results in this paper.